If you want to be super safe, don't claim exempt on the first job, because you no longer expect no tax liability at all for 2018 because of the new job. That's one of the prerequisites for claiming EXEMPT, that you expect to have no liability for 2018, and you didn't have any liability for 2017.
It's not vital that you do this, however, because if you really are making only $400 or less at this job the effect on your 2018 taxes will be minimal if not zero. But it's good that you are considering this, because so many taxpayers don't realize that all of your W-2s from all of your jobs get added together at the end of the year to compute your taxable income, and to figure your tax for the year. It doesn't make a difference whether you work the jobs at the same time or sequentially.
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